I have found a number of series where I can prove it case by case, but am having some trouble generalizing. In addition, if the series I am having an embarrassing time showing the result at all.

[snip]

If then

.

since the harmonic series diverges to infinity a heck of a lot slower than n does (I know that's not a solid technical argument but I'm sure it can be made so easily ..... is the glib 'easily' business the sticking point?)

Thank you for the effort, but this statement is incorrect. Perhaps the mistake in terminology Plato pointed out is a problem. So let me restate the original problem in more correct terms:

Suppose a sequence has the property as .

(That is to say has a limit point at infinity.)

Prove that

Since we are talking about a sequence here we cannot say that for all n, we are saying that the sequence approaches a limit point , as n approaches infinity. That is to say there is only one sequence, not an infinite number of sequences tending toward the same limit.

I proved this for my analysis class last year . It usues real numbers, but you can easily generalize it. Here.

I am somewhat confused almost from the beginning by the line

Is this again a case of my mistaken wording? If you define to be the nth partial sum of a series, this statement is quite understandable and falls out naturally. (Hopefully by now it is clear that this is not what I meant.)

Is this again a case of my mistaken wording? If you define to be the nth partial sum of a series, this statement is quite understandable and falls out naturally. (Hopefully by now it is clear that this is not what I meant.)